Dynkin system

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set $\Omega$ satisfying a set of axioms weaker than those of 𝜎-algebra. Dynkin systems are sometimes referred to as 𝜆-systems (Dynkin himself used this term) or d-system. These set families have applications in measure theory and probability.

A major application of 𝜆-systems is the π-𝜆 theorem, see below.

Definition

Let $\Omega$ be a nonempty set, and let $D$ be a collection of subsets of $\Omega$ (that is, $D$ is a subset of the power set of $\Omega$). Then $D$ is a Dynkin system if

1. $\Omega \in D;$
2. $D$ is closed under complements of subsets in supersets: if $A,B\in D$ and $A\subseteq B,$ then $B\setminus A\in D;$
3. $D$ is closed under countable increasing unions: if $A_1\subseteq A_2\subseteq A_3\subseteq \cdots$ is an increasing sequence of sets in $D$ then $\bigcup _{n=1}^{\mathrm{\infty }}{A}_n\in D.$

It is easy to check that any Dynkin system $D$ satisfies:

4. $\varnothing \in D;$
5. $D$ is closed under complements in $\Omega$: if $A\in D,$ then $\Omega \setminus A\in D;$
   • Taking $A:=\Omega$ shows that $\varnothing \in D.$
6. $D$ is closed under countable unions of pairwise disjoint sets: if $A_1,A_2,A_3,\dots$ is a sequence of pairwise disjoint sets in $D$ (meaning that $A_i\cap A_j=\varnothing$ for all $i\ne j$) then $\bigcup _{n=1}^{\mathrm{\infty }}{A}_n\in D.$
   • To be clear, this property also holds for finite sequences $A_1,\dots ,A_n$ of pairwise disjoint sets (by letting $A_i:=\varnothing$ for all $i>n$).

Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class. For this reason, a small group of authors have adopted conditions 4-6 to define a Dynkin system.

An important fact is that any Dynkin system that is also a π-system (that is, closed under finite intersections) is a 𝜎-algebra. This can be verified by noting that conditions 2 and 3 together with closure under finite intersections imply closure under finite unions, which in turn implies closure under countable unions.

Given any collection $𝒥$ of subsets of $\Omega ,$ there exists a unique Dynkin system denoted $D\{𝒥\}$ which is minimal with respect to containing $𝒥.$ That is, if $\tilde{D}$ is any Dynkin system containing $𝒥,$ then $D\{𝒥\}\subseteq \tilde{D}.$ $D\{𝒥\}$ is called the Dynkin system generated by $𝒥.$ For instance, $D\{\varnothing \}=\{\varnothing ,\Omega \}.$ For another example, let $\Omega =\{1,2,3,4\}$ and $𝒥=\{1\}$; then $D\{𝒥\}=\{\varnothing ,\{1\},\{2,3,4\},\Omega \}.$

Sierpiński–Dynkin's π-λ theorem

Sierpiński-Dynkin's π-𝜆 theorem: If $P$ is a π-system and $D$ is a Dynkin system with $P\subseteq D,$ then $\sigma \{P\}\subseteq D.$

In other words, the 𝜎-algebra generated by $P$ is contained in $D.$ Thus a Dynkin system contains a π-system if and only if it contains the 𝜎-algebra generated by that π-system.

One application of Sierpiński-Dynkin's π-𝜆 theorem is the uniqueness of a measure that evaluates the length of an interval (known as the Lebesgue measure):

Let $(\Omega ,ℬ,\ell )$ be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let $m$ be another measure on $\Omega$ satisfying $m[(a,b)]=b-a,$ and let $D$ be the family of sets $S$ such that $m[S]=\ell [S].$ Let $I:=\{(a,b),[a,b),(a,b],[a,b]:0<a\le b<1\},$ and observe that $I$ is closed under finite intersections, that $I\subseteq D,$ and that $ℬ$ is the 𝜎-algebra generated by $I.$ It may be shown that $D$ satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's π-𝜆 Theorem it follows that $D$ in fact includes all of $ℬ$, which is equivalent to showing that the Lebesgue measure is unique on $ℬ$.

Application to probability distributions

The π-𝜆 theorem motivates the common definition of the probability distribution of a random variable $X:(\Omega ,ℱ,\mathrm{P})\to ℝ$ in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable is defined as $$F_X(a)=\mathrm{P}[X\le a],a\in ℝ,$$ whereas the seemingly more general law of the variable is the probability measure $${ℒ}_X(B)=\mathrm{P}[X^{-1}(B)]\text{ for all }B\in ℬ(ℝ),$$ where $ℬ(ℝ)$ is the Borel 𝜎-algebra. The random variables $X:(\Omega ,ℱ,\mathrm{P})\to ℝ$ and $Y:(\tilde{\Omega },\widetilde{ℱ},\tilde{\mathrm{P}})\to ℝ$ (on two possibly different probability spaces) are equal in distribution (or law), denoted by $X\stackrel{𝒟}{=}Y,$ if they have the same cumulative distribution functions; that is, if $F_X=F_Y.$ The motivation for the definition stems from the observation that if $F_X=F_Y,$ then that is exactly to say that ${ℒ}_X$ and ${ℒ}_Y$ agree on the π-system $\{(-\mathrm{\infty },a]:a\in ℝ\}$ which generates $ℬ(ℝ),$ and so by the example above: ${ℒ}_X={ℒ}_Y.$

A similar result holds for the joint distribution of a random vector. For example, suppose $X$ and $Y$ are two random variables defined on the same probability space $(\Omega ,ℱ,\mathrm{P}),$ with respectively generated π-systems ${ℐ}_X$ and ${ℐ}_Y.$ The joint cumulative distribution function of $(X,Y)$ is $$F_{X,Y}(a,b)=\mathrm{P}[X\le a,Y\le b]=\mathrm{P}[X^{-1}((-\mathrm{\infty },a])\cap Y^{-1}((-\mathrm{\infty },b])],\text{ for all }a,b\in ℝ.$$

However, $A=X^{-1}((-\mathrm{\infty },a])\in {ℐ}_X$ and $B=Y^{-1}((-\mathrm{\infty },b])\in {ℐ}_Y.$ Because $${ℐ}_{X,Y}=\{A\cap B:A\in {ℐ}_X,\text{ and }B\in {ℐ}_Y\}$$ is a π-system generated by the random pair $(X,Y),$ the π-𝜆 theorem is used to show that the joint cumulative distribution function suffices to determine the joint law of $(X,Y).$ In other words, $(X,Y)$ and $(W,Z)$ have the same distribution if and only if they have the same joint cumulative distribution function.

In the theory of stochastic processes, two processes $(X_t{)}_{t\in T},(Y_t{)}_{t\in T}$ are known to be equal in distribution if and only if they agree on all finite-dimensional distributions; that is, for all $t_1,\dots ,t_n\in T,n\in \mathbb{N},$ $$(X_{t_1},\dots ,X_{t_n})\stackrel{𝒟}{=}(Y_{t_1},\dots ,Y_{t_n}).$$

The proof of this is another application of the π-𝜆 theorem.

See also

• Algebra of sets – Identities and relationships involving sets
• δ-ring – Ring closed under countable intersections
• Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
• Monotone class – Measure theory and probability theoremPages displaying short descriptions of redirect targets
• π-system – Family of sets closed under intersection
• Ring of sets – Family closed under unions and relative complements
• σ-algebra – Algebraic structure of set algebra
• 𝜎-ideal – Family closed under subsets and countable unions
• 𝜎-ring – Family of sets closed under countable unions

Further reading

This article incorporates material from Dynkin system on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Families $ℱ$ of sets over $\Omega$
Is necessarily true of $ℱ:$ or, is $ℱ$ closed under:	Directed by $\supseteq$	$A\cap B$	$A\cup B$	$B\setminus A$	$\Omega \setminus A$	$A_1\cap A_2\cap \cdots$	$A_1\cup A_2\cup \cdots$	$\Omega \in ℱ$	$\varnothing \in ℱ$	F.I.P.
π-system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if $A_i\searrow$	only if $A_i\nearrow$
𝜆-system (Dynkin System)				only if $A\subseteq B$			only if $A_i\nearrow$ or they are disjoint			Never
Ring (Order theory)
Ring (Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra (Field)										Never
𝜎-Algebra (𝜎-Field)										Never
Filter
Proper filter				Never	Never				Never
Prefilter (Filter base)
Filter subbase
Open Topology							(even arbitrary $\cup$)			Never
Closed Topology						(even arbitrary $\cap$)				Never
Is necessarily true of $ℱ:$ or, is $ℱ$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in $\Omega$	countable intersections	countable unions	contains $\Omega$	contains $\varnothing$	Finite Intersection Property
Additionally, a semiring is a π-system where every complement $B\setminus A$ is equal to a finite disjoint union of sets in $ℱ.$ A semialgebra is a semiring where every complement $\Omega \setminus A$ is equal to a finite disjoint union of sets in $ℱ.$ $A,B,A_1,A_2,\dots$ are arbitrary elements of $ℱ$ and it is assumed that $ℱ\ne \varnothing .$